1

Numerical simulation of electrically-driven flows using OpenFOAM

Francisco Pimenta a and Manuel A. Alves b

CEFT, Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto

Frias, 4200-465 Porto, Portugal

In this work, we study the implementation of electrically-driven flow (EDF)

models in the finite-volume framework of OpenFOAM®. The Poisson-

Nernst-Planck model is used for the transport of charged species and it is

coupled to the Navier-Stokes equations, governing the fluid flow. In addition,

the Poisson-Boltzmann and Debye-Hückel models are also implemented. The

discretization method and the boundary conditions are carefully handled in

order to ensure conservativeness of ionic species in generic meshes and

second-order accuracy in space and time. The applicability of the developed

solver is illustrated in two relevant EDFs: induced-charge electroosmosis

around a conducting cylinder and charge transport across an ion-selective

membrane. The solver developed in the present work, freely available as

open-source, can be a valuable and versatile tool in the investigation of

generic electrically-driven flows.

_________________________________

a fpimenta@fe.up.pt b mmalves@fe.up.pt

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I. INTRODUCTION

The interaction between fluids and electric fields at the micro- and nano-scales is at

the origin of several phenomena with increasing interest. These interactions are frequently

grouped in two classes, electrohydrodynamics (EHD) and electrokinetics (EK) 1, 2, that

we collectively call electrically-driven flows (EDFs) throughout this work. While EHD

is characterized by the accumulation of net-charge at an interface, the distinguishing

feature of EK is the formation of neutral electric double layers (EDLs) 1. EK and EHD

play an important role in fuel cells 3, electrodyalisis 4, separation techniques 5, 6, fluid

pumping 7, 8 and mixing 8, 9 in microfluidic systems, among other applications.

Understanding EK and EHD phenomena through theoretical models has been a

challenge pursued over the last decades. Nevertheless, some questions remain

unanswered and gaps still exist in the very basic theory 1, 10. As in other areas of physics,

this is both due to the failure of the theory in capturing all the events involved – owing to

simplifications or simply because they are unknown –, and to difficulties in extracting

solutions from the available models. Regarding this last issue, closed form solutions to

the system of equations that typically arises in EDFs are often difficult to be obtained,

even for simple geometries. Matched asymptotes 11, 12 and numerical methods 13-17 are

among the approaches frequently adopted to solve the resulting system of equations,

where the latter are particularly suitable to simulate EDFs in complex geometries.

Finite-differences 13, 18-25, finite-elements 13, 26-31, finite-volumes 16, 32, 33 , 34-43 and

lattice Boltzmann 15, 44-46 are numerical methods that have been successfully used to

simulate EDFs. In what concerns available software packages, COMSOL Multiphysics®

(https://www.comsol.com/), based on the finite-element method, has been a popular

choice for this purpose 13, 26-29. In the range of open-source packages, some works using

Gerris 32, 33 and OpenFOAM® 34-39, 47-49 can be found in the literature, but none of these

packages consistently offers a wide-range of ready-to-use models for EDFs, although

both allow the user to build its own models. In this work, we use the OpenFOAM®

package due to the following features: devoid of paid licenses, handling of generic

polyhedral grids, parallel computation capability, possibility of creating/changing the

source code, ease of integration between modules (rheological models, multiphase

models, etc.) and wide acceptance among both the academic and industrial communities.

Concerning the simulation of EDFs with OpenFOAM®, Nandigana and Aluru 37

coupled the Poisson-Nernst-Planck equations to the Navier-Stokes equations to study the

I-V curve in a micro-nanochannel. An area-averaged multi-ion transport model has then

3

been derived to study similar problems 38, 39. The Poisson-Nernst-Planck model, coupled

to the Stokes equations, has also been used to study the flow inside a nanopore in steady

conditions 47-49. Zografos et al 36 optimized contraction-expansion microchannels to

generate homogeneous extensional electroosmotic flows using the Debye-Hückel model,

a simplification of the Poisson-Nernst-Planck model. In the range of two-phase EDFs,

Lima and d’Ávila 34 implemented a leaky dielectric model to study the deformation of

viscoelastic droplets under the action of electric fields. Roghair et al 35 used a similar

model to analyze electrowetting of Newtonian fluids, where both the fluid and solid

domains were numerically simulated in a coupled way. To the best of our knowledge,

only the solver used in the work of Roghair et al 35 has been made available to the public,

which, however, only implements the leaky dielectric model for two-phase EDFs of

Newtonian fluids.

This work reports the numerical implementation of models to simulate single-phase

EDFs of Newtonian fluids, using the OpenFOAM® toolbox. The models are implemented

on the top of rheoTool (https://github.com/fppimenta/rheoTool), an open-source toolbox

for the simulation of Generalized Newtonian/viscoelastic fluid flows using OpenFOAM®,

which is now able to simulate both pressure- and electrically-driven flows (individually

or mixed). We restrict this work to the analysis of electrokinetic problems, using the

Poisson-Nernst-Planck, Poisson-Boltzmann and Debye-Hückel models, but other models

(slip models, Ohmic model, among others) and additional cases can be found in rheoTool.

The applicability of the solver is further demonstrated in the simulation of two particular

EDFs: induced-charge electroosmosis (ICEO) around a conducting cylinder and charge

transport across an ion-selective membrane. These cases are also used to assess the

accuracy and stability of the numerical algorithm, and the results obtained can be used

for benchmark purposes. Overall, this work is an attempt to increase the availability of

general-purpose open-source solvers with built-in EDF models, for fluids of general

rheology.

The remainder of this paper is organized as follows: Section II presents the governing

equations and Section III describes the details of their numerical implementation in the

finite-volume framework of OpenFOAM®. The performance of the numerical method in

the application cases is discussed in Section IV. Finally, Section V presents the main

conclusions and perspectives for future works.

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II. GOVERNING EQUATIONS

In the simulation of EDFs, two different, but coupled, components can be identified:

the hydrodynamic component, represented by the continuity and momentum equations;

the electric component, usually embodied by the Poisson-Nernst-Planck equations for the

transport of charged species in dilute electrolytes. Both components are addressed in the

next sections.

A. Hydrodynamics: continuity and momentum equations

Consider the transient, incompressible, isothermal, single-phase, laminar flow of a

Newtonian fluid under the action of an electric force. The continuity (Eq. 1) and

momentum (Eq. 2) equations are

0 u (1)

E

2fuuu

u

p

t (2)

where u is the velocity vector, t is the time, p is the pressure, fE represents the electric

force per unit volume, ρ is the fluid density and η is the viscosity.

The coupling between hydrodynamics and the electric force is ensured by the term fE.

Ignoring magnetic effects, this term can be derived from the electrostatic Maxwell stress

tensor,

I

EEEσ

2

2

, by taking its divergence

2

2

EE

EEσf (3)

where ε is the electric permittivity of the fluid, E is the electric field and E is the charge

density. The electric permittivity of the fluid is considered constant in all the problems

addressed in this work, thus Ef EE in those cases. While the electric field is

generically defined by the negative gradient of the electric potential, the computation of

E is model-dependent, as we show next.

B. Poisson-Nernst-Planck equations

The transport of charged species (ions) in a low-ionic strength electrolyte, under the

action of an electric field, can be described by the Nernst-Planck equation

iiiiii cΨcDct

c

u (4)

5

where ci is the molar concentration, Di is the diffusion coefficient, i is the electrical

mobility and Ψ is the electric potential, with ΨE in electrostatics. The index i

represents each individual species in the electrolyte. The terms in the left hand-side of Eq.

(4) represent the material derivative of ci, the first term in the right hand-side is the flux

of ci due to diffusion and the last term represents the transport of ci by the action of the

electric field, also known as electromigration. This electromigration term is formally

similar to a convective term, where an electromigration velocity (or flux, after

discretization) can be identified, Ψ iiM, u . All the electrolytes considered in this

work follow the Nernst-Einstein relation, kT

ezD i

ii , where zi is the charge valence, e

is the elementary charge, k is the Boltzmann constant and T is the absolute temperature.

However, for the sake of generality we will keep the notation for a generic i , since the

Nernst-Einstein relation does not apply for several polyelectrolytes, such as for example

DNA molecules 50.

The electric potential distribution in a given domain can be computed from Gauss’

law (ignoring polarization)

E Ψ (5)

with the charge density defined as

m

czF1i

iiE (6)

where F represents Faraday’s constant and m is the number of different ionic species. Eqs.

(1)–(6) form the basic theory used to simulate most of EDFs. However, the electric

component of this set of partial differential equations is frequently simplified or

transformed in order to avoid numerical issues, as for example the numerical stiffness

arising from the significantly different length and time scales which may co-exist, or

simply to enable the derivation of closed-form solutions. Two of these simplified models

are presented next. Hereafter, the Poisson-Nernst-Planck model, defined by Eqs. (4)–(6),

is abbreviated as PNP.

C. Poisson-Boltzmann model

The Poisson-Boltzmann model is a widely adopted simplification of the PNP model,

and is based on the assumption that the ionic species follow a Boltzmann distribution 44,

51, 52,

6

0

i

ii,0i exp

Dcc (7)

where ci,0 is a reference concentration of species i for which 0 , and represents

the intrinsic electric potential, as described later in Section 3.1. Eq. (7) can also be seen

as the result of integrating the Nernst-Planck equation (Eq. 4) to a reference point (where

ci = ci,0 and 0 ), assuming a zero material derivative (steady-state and negligible

transport of ions by convection). Variable Ψ has been replaced by in order to ease the

definition of 0 in a generic situation where an externally applied electric potential co-

exists with an intrinsic potential. This simplified model has a restricted applicability

comparing to the generic PNP model, mainly due to the assumption of Boltzmann

equilibrium to a fixed reference point. More details on these restrictions can be found

elsewhere 52 but, roughly speaking, the Poisson-Boltzmann model is essentially valid for

steady-state problems with negligible charge transport, non-overlapping EDLs and low

intrinsic potentials.

We will assume 00 henceforth, which is equivalent to consider electroneutrality

in the reference point, usually the bulk solution. This assumption imposes no further

restrictions other than the ones previously mentioned (different pairs 0,i0 ,c would

equally define Eq. 7), and is only taken to remove variable 0 from the equation. When

Eq. (7) is inserted in Eq. (6), the charge density of the so-called Poisson-Boltzmann model

is obtained

m

DczF

1i i

ii,0iE exp

(8)

Eqs. (5) and (8) define the Poisson-Boltzmann model, hereafter abbreviated as PB,

where the electric component is now decoupled from the hydrodynamics (but not the

opposite). In addition, the only unknown regarding the electric component of the system

is the electric potential, which is computed from Eqs. (5) and (8).

D. Debye-Hückel model

The PB model can be further simplified to the so-called Debye-Hückel model in the

limit of low electric potentials, 1i

i

D. Under this approximation, the charge density

can be obtained by expanding the exponential term of Eq. (8) in a Taylor series up to the

second term

7

m

DczF

1i i

ii,0iE 1

(9)

Eqs. (5) and (9) define the Debye-Hückel model, hereafter abbreviated as DH. Again,

the electric component is also decoupled from the hydrodynamics.

III. NUMERICAL METHOD

The numerical discretization of the governing equations in the finite-volume

framework follows the standard procedure available in OpenFOAM®, described

elsewhere for related partial differential equations, e.g. in Moukalled et al 53 and Pimenta

and Alves 54. This methodology is applied in a consistent operator-basis for collocated

grids. For the sake of conciseness, the discretization procedure for standard operators

(time-derivatives, convective terms, etc.) is not presented here. In this section, we restrict

our analysis to the particularities of EDFs.

A. Splitting the electric potential into external and intrinsic electric potentials

When solving EDF problems with the simplified models (PB and DH), it is often

convenient to decompose the electric potential in two variables, Ext Ψ , where Ext

is the electric potential externally imposed and is the electric potential that exists

intrinsically in the channel, commonly associated with the EDL formation 51. Under this

approach, Gauss’ law is also decomposed into two equations

E

Ext

E

0

Ψ (10)

solved independently and sequentially for each variable. Following this strategy, it is also

a common practice to only consider the contribution from Ext in the computation of the

electric field entering the electric force term of the momentum equation (Eq. 3), thus

ExtEE f , as explained in Kyoungjin et al 51. This is mainly to avoid the development

of a large pressure gradient near the walls, due to the normal gradient of in these

regions – this contribution can be canceled in the momentum equation by considering that

it is balanced by an opposing pressure gradient, assuming that it does not influence the

flow 51. Such transformation can be particularly important (advantageous) in the case of

viscoelastic fluids, where the development of large unbalanced pressure gradients may

artificially trigger flow instabilities.

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B. Wall boundary conditions in EDFs

Several materials used in EDFs at the micro- and nano-scales are impermeable and

electrically insulating, as for example glass and polydimethylsiloxane (PDMS). In contact

with an electrolyte, these materials develop electric double layers at the solid/liquid

interface. The boundary conditions usually assigned to such surfaces, considering the

PNP model, are a null velocity, a no-flux condition for ions and a specified electric

potential or specified surface-charge (constant or variable over the surface).

The no-flux condition at the wall leads to (assuming no-penetration on the wall)

00 ffiiiifi, SΨccDF (11)

where Sf is the wall-normal vector, whose magnitude represents the area of face f lying

on the wall. Eq. (11) is formally a Robin-type boundary condition for ci. There are several

options to convert Eq. (11) into a Neumann or Dirichlet boundary condition, for the ease

of implementation, and three of these methods are as follows

nn(III)

(II)

nn(I)

fi

ifi,fiPf

i

iPi,fi,

Pf

i

i

Pi,

fi,

fi

ifi,fi

andexp

1

ΨD

ccΨΨD

cc

ΨΨD

cc

ΨD

cc

with ci,P representing the value of ci evaluated at the center of the cell owning the boundary

face where ci,f is to be computed (subscripts P and f represent the cell and the face,

respectively), and f

f

S

Sn is the unitary vector, normal to face f. Method (I) results

directly from Eq. (11) by simply isolating each term in a different side of the equation.

Method (II) is obtained from method (I) after discretizing the gradients on the boundary

and isolating all the ci,f terms. Method (III) derives an expression for ci,f based on the

analytical integration of Eq. (11) between the cell and the face, which naturally results in

an exponential variation of ci near the boundary. This exponential behavior is preserved

when computing the gradient of ci by using the analytical expression for ci,f , instead of

taking

Pi,fi,

fi

ccc

n , with δ being the distance between the cell center and face f.

This last approach would generically linearize the exponential variation, leading to higher

errors in coarse grids.

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The three methods above are strictly similar and effective in imposing Fi,f = 0. Indeed,

all satisfy equally Eq. (11) at the discrete level, which states that the diffusive and

electromigration fluxes cancel each other at the boundary. This could be otherwise

accomplished at the matrix construction stage by simply ignoring the contributions (at the

no-flux boundary) from the diffusive and electromigration fluxes to the source vector,

thus avoiding the need to compute ci,f or its gradient at the wall. In practice, the three

methods above can only be distinguished based on the accuracy of the ci,f value retrieved

when this variable is used for post-processing purposes, or when it is used in some high-

order methods to increase the accuracy in non-orthogonal grids. In addition, the value of

ci,f is also needed to compute limiters when using high-resolution schemes for the

convective term of Eq. (4). Based on these criteria, method (III) was seen to be more

accurate and less mesh-dependent in non-orthogonal grids. Thus, method (III) was used

throughout this work. It should be noted that when using this method to evaluate ci,f , the

value of ci,P from the previous time-step/iteration is considered, resulting in an explicit

boundary condition (an implicit implementation would be also a possibility).

Regarding the electric potential at the wall, as aforementioned the two common

choices are either imposing the electric potential, or defining the surface charge density

(σ), which can be related with the electric potential,

n

fΨ . For the PNP model,

imposing the electric potential at the wall is a less natural boundary condition and it is

also more complex to implement for walls of arbitrary shape, if a single electric potential

variable is being used. A common workaround to this issue is to also decompose the

electric potential in the PNP model.

When using the simplified PB and DH models, only the boundary conditions for the

two electric potentials are required, since they are the only electric-related variables being

solved. Considering the decomposition of the electric potential discussed in the previous

section, the externally applied electric field is tangential to an insulating wall, while the

intrinsic potential may be related with the zeta-potential (ξ) or with the surface charge

density,

potential) appliedy (externall 0

potential) (intrinsic or

fExt

ff

n

n

(12)

In order to complete the discussion regarding the boundary conditions assigned at a

wall, the pressure variable remains to be analyzed. A generic boundary condition can be

10

derived for this variable from the pressure (continuity) equation. Indeed, recalling the

SIMPLEC method described in our previous work 54, the velocity in a generic cell P can

be written as

1P

P

P

PH

a

p

aB

Hu (13)

which is the equation applied to correct the velocity after computing the pressure, being

also used in the formulation of the pressure equation. In Eq. (13), aP represents the

diagonal coefficients of the momentum equation and H1 is the negative sum of all the off-

diagonal coefficients. H is a vector containing the negative sum of the off-diagonal

coefficients multiplied by the respective velocity and it also takes into account the source

term contributions to the momentum equation, except the pressure gradient. The term

P

P1P

*1

H

1p

aa

B is specific from the SIMPLEC algorithm, and more details

can be found in Pimenta and Alves 54. When Eq. (13) is interpolated to the faces of the

domain, imposing a no-penetration condition at the wall ( 0f nu ) results in the

following boundary condition for the pressure

nBH

n

f

P

f1Pf

Ha

ap (14)

where both the aP and H1 coefficients are from the cell owning the boundary face (only

operators H and B are directly evaluated at the face). Eq. (14) has the advantage of being

independent of the form taken by the momentum equation, since the contributions from

all its terms are condensed in generic operators. Physically, Eq. (14) establishes a balance

between the pressure and the remaining sources of momentum at the wall, in our case,

viscous and electric stresses. A common approach in generic incompressible flows is to

use 0f

np , the so-called zero-gradient approach for pressure. This approximation

does not violate continuity, but the pressure gradient at the boundary will not necessarily

balance the remaining forces in the momentum equation, which can be of considerable

magnitude in the case of EDFs, unless the normal electric stresses are artificially removed

(balanced), as described in the previous section for the simplified models.

C. Discretization of the electromigration term

The electromigration term of the Nernst-Planck equation (Eq. 4) can be computed in

different ways, which despite being equivalent analytically, might differ at the discrete

level. One option is to split the term according to the properties of the divergence operator,

11

ΨcΨccΨ iiiii (15)

While the second term of Eq. (15) is likely to be discretized explicitly, the first term can

be discretized semi-implicitly using Eqs. (5) and (6).

A second option is to handle the electromigration as a standard convective term,

allowing for an implicit discretization. In this approach, the electromigration flux,

ffifiM, S ΨF , needs to be evaluated consistently on the cell faces in order to

balance the remaining fluxes. This can be achieved by computing f

Ψ , the negative of

the electric field at cell faces, using the electric potential values straddling each face,

instead of interpolating the gradients evaluated at the cell centers. The concentration

variable, ci, is then interpolated to face centers using an adequate scheme.

A third option is to simply consider the electromigration term as the Laplacian of the

electric potential, with a variable coefficient iic that can be linearly interpolated from

cell centers to face centers. The implementation of this method requires an explicit

evaluation of the electromigration term.

The last two methods were tested in the cases addressed in this work and no significant

differences were observed between them in what respects accuracy and stability. Thus,

the last method has been used due to its lower computational cost. Importantly, the no-

flux boundary condition should be consistent with the discretization of the

electromigration term, such that the fluxes exactly cancel out at the boundary.

D. Linearization of the exponential terms in the Poisson-Boltzmann equation

In order to improve the numerical stability of the PB model, the exponential source

term included in Eq. (8) is linearized using a Taylor expansion up to the second term 55.

In its original form, the electric potential for this model is computed from

m

a

b

DczF

1i

i

i

i

ii,0i exp

(16)

where the two variables ai and bi were introduced for the ease of notation in what follows.

After linearization of the source term on the right hand-side of Eq. (16), the following

equation is obtained

m

1i

ii

m

1i

i

m

1i

ii

****baFaFbaF (17)

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where the terms with a star are evaluated explicitly. When the steady solution of Eq. (17)

is reached, the second term in both sides of the equation exactly cancel each other and the

original equation (Eq. 16) is recovered. In transient computations, the explicitness can be

reduced by iterating Eq. (17) multiple times in order to update the terms with a star at

each time-step.

E. Discretization schemes

The convective terms are discretized with the CUBISTA high-resolution scheme 56,

following a deferred correction approach 54. Both the Laplacian and gradient terms were

discretized using central-differences. Overall, the algorithm is second-order accurate in

space.

The time-derivatives were discretized using the three-time level scheme 54, rendering

the algorithm also second-order accurate in time, as will be shown later.

All the terms of the momentum equation (Eq. 2), except the pressure gradient and the

electric contribution, are discretized implicitly. In the Nernst-Planck equation (Eq. 4),

only the electromigration term is accounted for explicitly, while in Poisson-type equations

(either for pressure or for the electric potential) only the Laplacian operator is discretized

implicitly, except in the Poisson equation for the DH model, where part of the E term

(Eq. 9) is also accounted for implicitly. Note that when mentioning implicit discretization,

this excludes the corrective terms stemming, for example, from the deferred correction of

convective terms (when using high-resolution schemes) or from the non-orthogonal

correction of the Laplacian operator 53.

F. Overview of the algorithm

We use the same background solving sequence described in detail in Pimenta and

Alves 54, which has been modified here to include the electric-related steps. In this

segregated scheme, the coupling between the pressure and velocity fields is ensured by

the SIMPLEC algorithm and an inner-iteration loop is used to reduce the explicitness of

the method, and to increase its accuracy and stability. More details can be found in

Pimenta and Alves 54. Briefly, the sequence adopted in this work to solve EDFs consists

of the following steps (noting that Ψ should be replaced by Ext and for the PB and

DH models, which do not include ci as a computed variable):

1- Initialize the fields { p, u, Ψ, ci }0 and time (t = 0)

2- Enter the time loop (t = Δt)

2.1- Enter the inner iterations loop (j = 0)

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2.1.1- Enter the electrokinetic coupling loop (k = 0)

2.1.1.1. Compute Ψ from Eq. (5) for the PNP model, or Eqs. (10) for PB and DH

models

2.1.1.2. Compute ci from the Nernst-Planck equation (Eq. 4) – skip this step for

PB and DH models

2.1.1.3. Increment the loop index (k = k + 1) and return to step 2.1.1.1 until the

pre-defined number of coupling iterations is reached (only one iteration

is needed for the PB and DH models)

2.1.2- Solve the momentum equation (Eq. 2)

2.1.3- Solve the pressure equation to enforce continuity (Eq. 1)

2.1.4- Increment the inner iteration index (j = j + 1) and return to step 2.1.1 until the

pre-defined number of inner iterations is reached

2.1.5- Set { p, u, Ψ, ci }t = { pj, uj, Ψj, ci,j}

2.2- Increment the time, t = t + Δt, and return to step 2.1 until the final time is reached

3- Stop the simulation and exit

From the above sequence, it is worth to mention the need of the so-called

electrokinetic coupling loop (step 2.1.1). This loop is required to guarantee second-order

accuracy in time for the PNP model, as will be shown later, and to enhance the coupling

between the electric potential and the ionic concentration, reducing the non-linearity

embodied by the electromigration term. This loop is not used with the PB and DH models,

since those issues do not arise for these models.

The sparse matrices resulting from the discretization procedure are typically solved

using a Pre-conditioned Bi-Conjugate Gradient (PBiCG) method coupled with a Diagonal

Incomplete-LU (DILU) pre-conditioner, for non-symmetric matrices, and a Geometric-

Algebraic Multi-Grid (GAMG) method coupled with a Diagonal Incomplete-Cholesky

(DIC) pre-conditioner, for symmetric matrices. The absolute tolerance for the sparse-

matrix solvers is typically set at 10-10.

IV. RESULTS AND DISCUSSION

In this section, we start by assessing the convergence rate of the PNP model, both in

space and time, before proceeding to the two selected application cases. We should note

that all the cases addressed in this section are for symmetric, binary electrolytes, although

the code developed is generic for any charge valence, diffusivity and number of species

– the analysis of such generic cases is left as a suggestion for future work.

14

A. Conservativeness, spatial and temporal accuracy of the PNP model: the 2D cavity

As shown in the previous section, several methods can be used to compute

numerically the PNP equations. Even though some methods are analytically equivalent,

they display different properties at the discrete level. The properties that we require for

our discretized PNP equations are second-order accuracy in space and time, and

conservation of the ionic species. Thus, it is convenient to first assess these points before

advancing to more complex applications.

The problem selected for this purpose is the 2D cavity presented by Mirzadeh et al 18,

which was also used by those authors to assess the conservation of ions by the PNP model

in adaptive quadtree grids. It consists of a closed domain where an imposed sinusoidal

variation of the electric potential along the walls generates a non-uniform distribution of

ions (charge density). Due to the simple geometry (a 2D square) and conditions used, the

2D cavity is a good case to assess convergence rates and conservativeness for the PNP

model. Here, we retrieve a new benchmark variable to this case, making it also affordable

to probe the spatial accuracy of a numerical method.

1. Problem description

The cavity geometry is simply a square domain with side length 2H (Fig. 1). Although

most of the tests were conducted using an orthogonal, structured-like mesh, some

computations were also performed in a non-orthogonal mesh composed of triangles, as

depicted in Fig. 1. In both types of meshes, the cells were compressed towards the domain

boundaries, where the EDL develops (for the orthogonal mesh, the cell at each corner of

the domain has a square shape). The minimum boundary edge size for the orthogonal

meshes ranged between H/800 in mesh M1, and H/4000 in mesh M6, and the range was

H/1200 in mesh M1N, and H/4600 in mesh M3N, for the non-orthogonal-meshes. The

spatial resolution of all the meshes can be found in Fig. 2b.

The cavity is initially filled with a symmetric, binary electrolyte at uniform

concentration c0, for which z+ = -z– = z and D+ = D– = D. The boundary conditions at the

cavity walls are no-flux for both ionic species and

1/ /sin

1/ /sin

1/ /sin

1/ /sin

a

a

a

a

HxHyV

HxHyV

HyHxV

HyHxV

Ψ

15

for the electric potential (this expression is slightly different from the one presented by

Mirzadeh et al 18). The hydrodynamics is switched off in this problem, such that the ions

can only be transported by diffusion and electromigration.

FIG. 1. Square domain of the two-dimensional cavity, with side length 2H. The coordinate system is located

at the center of the domain. A zoomed view near one corner of the orthogonal and non-orthogonal meshes

is displayed next to the geometry.

The relevant dimensionless numbers in this problem are the dimensionless applied

voltage, T

a

a

~

V

VV , where

ez

kTV T is the thermal voltage, and the dimensionless Debye

parameter,

kT

FecHzH

0

2

D

2~ , where Fecz

kT

0

2D2

is the Debye length (an

estimate of the EDL thickness). We follow Mirzadeh et al 18 setting 5~

a V and 10~ .

The following dimensionless variables are used to display the results in this section:

2

~

H

tDt ,

H

yy ~ ,

H

xx ~ ,

T

~

V

ΨΨ and

0

E

)(~

c

cc .

2. Results

The contours of Ψ~

and E~ are plotted in Fig. 2a. The charge density is only non-

negligible near the walls, where it displays a negative or positive value in the regions of

positive or negative electric potential, respectively. The absolute charge density is

2H x

2H

y Orthogonal mesh

Non-orthogonal mesh

16

symmetric in relation to the two diagonals of the square domain, as also in relation to the

two Cartesian axes.

FIG. 2. (a) Contours of the electric potential (upper-diagonal) and absolute charge density (lower-diagonal)

in mesh M6. (b) Spatial convergence rate with mesh refinement. The solid line represents a power-law fit

to the numerical data obtained with the orthogonal mesh. In the x-axis we represent the minimum grid size

in the direction normal to the cavity boundaries (edge size of the cell adjacent to the boundaries). (c) Spatial

evolution of |ρE| along the x-direction, at y = 1, for different meshes (see mesh numbering in panel b). The

inset is a zoomed view of the profile near the peak region. (d) Temporal convergence rate for a different

number of electrokinetic coupling iterations. The solid line represent a power-law fit to the numerical data.

Mesh M4 was used to obtain these results (see mesh numbering in panel b).

We start assessing the spatial convergence rate using the peak values of E~ , at (x, y)

= (± H/2, ± H) ∨ (± H, ± H/2), which lie on the cavity boundaries (the local E~ results

from the no-flux boundary condition for the ions and from the imposed electric potential).

(c) (d)

(b) (a)

0

20

40

60

| ρE |

0

2

4

-4

-2

Ψ

0

10

20

30

40

50

60

0.0 0.2 0.4 0.6 0.8 1.0

| ρ E

|

x

M2 M4 M6

M1N M2N M3N

Err

or

Δx

Orthogonal mesh Non-Orthogonal mesh

10-4 10-3

101

100

10-1

2.06

M4

M6

M2N

M1NM1

M3N

M2

M3

M5

Err

or

Δt

1 iter. 2 iter. 3 iter. 4 iter.

10-410-8

10-2

10-7

10-3

10-4

10-5

10-6

10-7

10-6 10-5

1.02

1.99

17

We estimate the variable in one of the eight locations by fitting a parabola to the three

nearest face values lying on the boundary. The variable is computed in meshes of different

resolution and Richardson’s extrapolation to the limit is used to extrapolate the value for

an infinitesimal cell spacing (0E

~x

). Then, for each mesh resolution we compute the

error as 0EE

~~

xx

. The results obtained are displayed in Fig. 2b, where we can

confirm that the numerical method is second-order accurate in space. The non-orthogonal

mesh presents a higher error than the orthogonal one, and its spatial accuracy seems to be

approaching second-order for the two most refined meshes. The spatial variation of E~

on the boundaries is plotted in Fig. 2c for different mesh resolutions, showing the

convergence of the profiles (dependency on the mesh resolution is more clear near the

peak region). Using the data from the orthogonal meshes, the Richardson’s extrapolated

value of E~ at (x, y) = (± H/2, ± H) ∨ (± H, ± H/2) is 62.467.

In order to estimate the convergence rate in the temporal dimension, the value of E~

was collected at t = 0.001, at the same position, i.e., (x, y) = (± H/2, ± H) ∨ (± H, ± H/2),

using different time-steps and keeping the same mesh resolution. The extrapolation of

that variable for an infinitesimal time-step was conducted as described above for the

spatial convergence test and, for each time-step, the error was computed as

0EE~~

tt . The results are shown in Fig. 2d for a different number of iterations

of the electrokinetic coupling loop (loop 2.1.1 in the algorithm described in section 3.6).

For a single iteration, the method is only first-order accurate, notwithstanding the second-

order accuracy of the discretization scheme employed for the time-derivatives. Second-

order accuracy is only recovered for two or more coupling iterations. A similar behavior

was described and explained in Karatay et al 13 for the PNP system of equations. We do

not observed a significant improvement of accuracy by increasing from two to three or

four iterations. Thus, two electrokinetic coupling iterations were typically used

throughout this work.

As a final test, we computed the average change in the concentration of positive and

negative ions for different time-steps, in both orthogonal and non-orthogonal meshes.

Since no-flux boundary conditions are assigned to all domain boundaries, the average

concentration of both ions must be conserved over time. For these specific tests, the

absolute tolerance of the sparse-matrix solver of the Nernst-Planck equations was reduced

18

to 10-14. The average concentration variation for each ionic species was quantified as

NC

1k

kki,

T

i 1~1~ VcV

c where VT is the total volume of the domain, Vk is the volume of

cell k and NC is the total number of cells in the domain. The sum is carried out at t = 2,

which is already in the steady-state regime of ci. For a fully conservative method, 0~i c

is expected. The range of time-steps tested spans two orders of magnitude, Δt ∈ [0.0001,

0.01], and the values are significantly higher than the ones used to assess the temporal

convergence rate, since higher losses are expected when using higher time-steps (Δt =

0.01 corresponds to one EDL charging time, λD2/D). The results obtained are presented in

Table I. The maximum loss observed was of O(10-11), which confirms that the numerical

method is conservative, both in orthogonal and non-orthogonal meshes. Furthermore,

there is no clear relation between the mass loss and the time-step used. In one hand, a

lower time-step leads to a more accurate evolution of the Nernst-Planck equations,

reducing the numerical error. On the other hand, a lower time-step also requires more

calculations until the steady-state is reached, which cumulatively deteriorates

conservativeness due to round-off errors and also due to the finite tolerance of the iterative

solvers.

TABLE I. Ionic species’ mass variation using different time-steps and different meshes. Two electrokinetic

coupling iterations were used for all the cases. See the definition of Δci in the text and mesh numbering in

Fig. 2b.

Δt Mesh M4 Mesh M2N

Δc+ Δc– Δc+ Δc–

0.01 2.69 x 10-12 6.03 x 10-12 3.12 x 10-12 1.02 x 10-12

0.005 6.35 x 10-12 2.47 x 10-13 6.19 x 10-13 4.24 x 10-12

0.001 6.00 x 10-13 2.56 x 10-12 5.20 x 10-13 2.21 x 10-12

0.0005 1.12 x 10-11 5.25 x 10-13 Not tested Not tested

0.0001 1.11 x 10-12 3.37 x 10-11 Not tested Not tested

B. Induced-charge electroosmosis (ICEO) around a conducting cylinder

The application case addressed in this section is the ICEO around a conducting

cylinder, driven by a DC electric field. This EDF arises, for example, when a conducting

(metallic) cylinder is placed in a DC (or AC) electric field. The potential induced on the

cylinder surface drives 4 symmetric counter-rotating vortices, at low to moderate induced

potentials 57. Theoretically, the velocity scales with the square of the electric field

19

magnitude (standard electroosmosis/electrophoresis only scale linearly). This feature,

associated with the possibility of using AC electric fields to drive a unidirectional flow,

attracted the interest of the scientific community on ICEO, and a number of practical

applications in microfluidics already exist 26, 58, 59. Thus, the numerical investigation of

ICEO is a relevant subject, even more due to the well-known failure of the basic theory

in capturing the velocity magnitude observed experimentally (e.g. 60, 61).

Sugioka 42 derived analytical expressions for the ICEO around a conducting cylinder,

considering approximate models for both low and high induced potentials. The author

also used a hybrid finite-element/finite-volume numerical method to assess the accuracy

of the theory developed. In what follows, we will compare our numerical results with the

numerical and analytical results of Sugioka 42. This case is suitable to test the accuracy

and stability of the solver in non-orthogonal meshes.

1. Problem description

The geometry consists of a 2D cylinder (radius R) centered in a square domain (Fig.

FIG. 3), filled with a symmetric, binary electrolyte (z+ = -z– = z and D+ = D– = D) at uniform

concentration, c0, and initially at rest. The electrodes, having symmetric electric potentials

(+V and –V), are placed on the north and south boundaries of the domain, spaced apart L

= 100R from each other, while the remaining boundaries are considered impermeable,

insulating walls. The size of the bounding domain was selected long enough in order to

guarantee that the solution is independent of this parameter. The domain was divided into

4 equal blocks in the azimuthal direction to build the mesh. For mesh M1, the mesh size

on the cylinder surface is ds = R/50 and dr = R/100 (320 cells in the azimuthal direction

and 80 cells in the radial direction). Mesh M2 was obtained from mesh M1 by doubling

the number of cells of each block, in each direction. The cells were uniformly distributed

in the azimuthal direction, and were compressed towards the cylinder surface in the radial

direction.

The following set of boundary conditions was considered for the PNP model:

Electrodes: Ψ = ±V, 0 np , u = 0, ci = c0;

Cylinder surface: Ψ = 0, np given by Eq. (14), u = 0, Fi = 0;

Insulating walls: 0 nΨ , 0 np , u = 0, Fi = 0.

20

Note that by imposing a fixed ionic concentration at the electrodes (a reservoir-like

condition), concentration polarization is not prone to happen at their surface, and we can

consider our case similar to the unbounded problem of Sugioka 42, regarding that point.

FIG. 3. Geometry used in the ICEO around a 2D conducting cylinder (drawing not to scale). The cylinder

has radius R and it is centered in a 100R square domain, composed of two electrodes having symmetric

electric potentials (north and south boundaries) and two lateral impermeable, insulating walls (west and

east boundaries). The cylinder has a fixed potential of 0 V.

The PB and DH models were also used in order to assess their accuracy against the

PNP model, regarding the steady-state solution. For the PB and DH models, the following

boundary conditions were used:

Electrodes: 0 , VExt , 0 np , u = 0;

Cylinder surface: Ext , 0Ext n , 0 np , u = 0;

Insulating walls: 0 n , 0Ext n , 0 np , u = 0.

This set of boundary conditions ensures 0Ext Ψ on the cylinder surface, i.e.,

the overall potential of the cylinder is kept constant and equal to its initial value.

A set of dimensionless numbers governing the ICEO are: dimensionless Debye

parameter,

kT

FecRzR

0

2

D

2~ ; Schmidt number, D

Sc

; electrohydrodynamic

L=100R θ

r

L=100R

+V

–V

E

2R

21

coupling constant, D

V

2

T ; dimensionless induced potential, T

~

V

ERV , where

ez

kTV T is the thermal voltage and

L

VE

2 is the applied electric field. To keep

consistency with Sugioka 42, creeping flow conditions were imposed by removing the

convective term of the momentum equation. In addition, χ = 0.47 and Sc = 103, while

both ~ and V~

were varied in the range 5–10 and 0.01–4, respectively. The following

dimensionless variables are used to present results: R

Rrd

~,

U

uu ~ and

0

E

)(~

c

cc , where

2REU is a characteristic velocity scale for ICEO 11. Note that

here we use characteristic scales which are different from Sugioka 42 to avoid that some

dimensionless numbers would depend on the size of the bounding domain.

2. Results

Fig. 4 presents radial profiles of the azimuthal velocity component, at θ = 45º, for

different combinations of parameters: V = {0.01; 4} and κ = {5; 10}. An excellent

agreement is observed in Fig. 4a,b between our numerical results in both meshes M1 and

M2 and the analytical solution of Sugioka 42, for the low induced voltage (V = 0.01). For

the higher induced voltage (V = 4), Fig. 4c,d show some discrepancies, although the

qualitative behavior is still captured. The differences at V = 4 are possibly a consequence

of the approximations at the basis of the high-voltage theory derived by Sugioka 42. In

general, our velocity profiles are in closer agreement with the analytical solution, than the

numerical results of Sugioka 42, particularly far from the cylinder surface. This is most

probably due to the longer domain used in the present work. The small domain used by

Sugioka 42 (L/R = 10) was seen to influence the results in the current work, worsening the

agreement with the analytical solution, which does not take into account wall effects.

The maximum azimuthal velocity component along the radial line θ = 45º is plotted

in Fig. 5a as a function of the dimensionless Debye parameter, for low and high voltages.

A good agreement is observed with the analytical and numerical results of Sugioka 42. As

the Debye parameter increases (the EDL thickness decreases), the dimensionless

maximum azimuthal velocity increases. On the other hand, for fixed κ = 10, the

dimensionless velocity decreases with increasing voltage, as illustrated in Fig. 5b for a

high-voltage range, and previously in the velocity profiles of Fig. 4. In experimental

22

ICEO, it is well known that the standard theory 11, from which U (velocity scale used to

normalize u) arises, overestimates the measured velocity 60, 61, and Fig. 5b shows that the

velocities computed numerically are also below the standard theory prediction, being

consistent with the experimental observations. The same theory predicts a quadratic

scaling of the velocity with the applied electric field, or equivalently U ∝ V 2, but the

numerical results suggest a scaling power lower than 2 (~1.6, see inset of Fig. 5b).

Davidson et al 20 also found a weaker dependence between the velocity at the poles of the

cylinder (θ = 0, 90) and the electric field, comparing to the standard theory prediction.

We should note, however, that the standard theory is essentially valid for κ >> 1 and V

<< 1 11.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5

uθ

d

Analytical (Ref.) Numerical (Ref.)

M1 (this work) M2 (this work)

V = 0.01

κ = 10

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

uθ

d

Analytical (Ref.) Numerical (Ref.)

M1 (this work) M2 (this work)

V = 0.01

κ = 5

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5

uθ

d

Analytical (Ref.) Numerical (Ref.)

M1 (this work) M2 (this work)

V = 4

κ = 10

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5

uθ

d

Analytical (Ref.) Numerical (Ref.)

M1 (this work) M2 (this work)

V = 4

κ = 5

(a) (b)

(c) (d)

23

FIG. 4. Azimuthal velocity component profiles along the radial coordinate, at θ = 45º: (a) V = 0.01 and κ =

10; (b) V = 0.01 and κ = 5; (c) V = 4 and κ = 10; (d) V = 4 and κ = 5. The analytical results for V = 0.01 and

V = 4 are from the low-voltage and high-voltage theory for an unbounded domain, respectively, obtained

from Sugioka 42. The reference numerical data is also from that work.

The characteristic vortices of the ICEO flow are shown in Fig. 6, together with the

charge density, for two different voltages V = {0.01; 4}, at κ = 10. While the velocity

magnitude profiles are symmetric in relation to the lines θ = 0-180º and θ = 90-270º, the

charge density is only symmetric relative to line θ = 0-180º, and it is anti-symmetric

relative to line θ = 90-270º. For the higher voltage (V = 4), a region of high velocity starts

to form at θ = 0, 180º and the region of non-zero charge density is compressed towards

the cylinder surface, which can only be seen in a zoomed view of Fig. 6.

FIG. 5. Maximum azimuthal velocity along the radial line θ = 45º for (a) different κ values, at V = 0.01 (LV

– low-voltage) and V = 4 (HV – high-voltage), and (b) different applied potentials, at κ = 10. In panel (a),

(a)

(b)

0

0.2

0.4

0.6

0.8

1

5 6 7 8 9 10

uθ,

max

κ

Analytical LV (Ref.) Analytical HV (Ref.)

Numerical LV (Ref.) Numerical HV (Ref.)

M2 LV (this work) M2 HV (this work)

V = 4

V = 0.01

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

uθ,

max

V

Analytical (Ref.) Numerical (Ref.)

M2 (this work)

κ = 10

0

1

10

100

1 10

Standard theoryNumerical (M2)

0.1

24

the analytical results for V = 0.01 and V = 4 are from the low-voltage and high-voltage theory for an

unbounded domain, respectively, as presented in Sugioka 42. In panel (b), the curve with the analytical

results is based on the high-voltage theory 42. The reference numerical data is also from that work. The plot

inset in panel (b) is a log-log representation of |uθ, max|/(D/R) as a function of V. The dashed line represents

the standard theory prediction: uθ=45,max = 2εRE2/η 11, also normalized in the same way.

Fig. 7 presents a comparison between the velocity profiles obtained with the PNP, PB

and DH models for high and low voltages. At low voltages, the two simplified models

agree well with the Poisson-Nernst-Planck model, but a significant deviation is observed

for the higher voltage, for which the assumptions at the basis of the two simplified models

(essentially the assumption of Boltzmann equilibrium and V << 1) do not hold. Thus,

using those two simplified models seems to be an acceptable approach at low voltages,

since the accuracy is not compromised and the computations are usually faster (higher

time-steps can be used due to the lower numerical stiffness and the CPU time per iteration

is also smaller). This is important, for example, in shape optimization problems, where

the steady-state solution has to be computed for each candidate geometry 36, and typically

hundreds of geometries are tested until the optimal design is found, thus making the

computational speed a key factor in these applications.

FIG. 6. Velocity contours with superimposed flow streamlines and charge density contours for V = 0.01

(top panels) and V = 4 (bottom panels). Each plot represents a different quadrant of the whole domain. All

the results are for κ = 10 and were computed in mesh M2.

75

50

25

0

-25

-50

-75

ρE |u|

0.4

0.3

0.2

0.1

0

ρE

0.04

0.02

0

-0.02

-0.04

|u|

0.8

0.6

0.4

0.2

0

V =

0.0

1

V =

4

θ

E

25

FIG. 7. Azimuthal velocity component profiles along the radial coordinate, at θ = 45º, for (a) V = 0.01 and

(b) V = 4, at κ = 10 (mesh M2), using different EDF models (PNP – Poisson-Nernst-Planck model; PB –

Poisson-Boltzmann model; DH – Debye-Hückel model). The analytical results for V = 0.01 and V = 4 stem

from the low-voltage and high-voltage theory in an unbounded domain, respectively, presented in Sugioka 42.

C. Charge transport across an ion-selective membrane

When an electric potential difference is applied over an ion-selective membrane, for

example in electrodialysis, the experimental I-V curve displays three regimes for

increasing V 62, 63: (i) Ohmic regime – I scales linearly with V; (ii) limiting regime – the

rate of increase of I with V decays and I approaches an asymptotic value due to the

diffusion-limited transport of ions; (iii) overlimiting regime – I increases again with V,

although not necessarily in a linear way. The existence of an overlimiting regime has been

commonly attributed to electroconvection – vortices form near the membrane and the

resulting advection overcomes the diffusion-limited transport of ions. This has been

observed experimentally 62-64 and predicted numerically 13, 21-25, 43, 65, 66, though direct

numerical simulations (DNS) of this problem are still relatively recent and only a few are

for 3D geometries 23, 25.

In this example, we use 2D DNS to obtain the I-V curve (or, equivalently, the J-V

curve, with J being the current density) for an ion-selective membrane. This last

application case is suitable to test the robustness and accuracy of the solver under the

chaotic flow conditions developed at high voltages (electroconvective instabilities).

1. Problem description

The 2D reservoir considered in the present work (Fig. 8) has the same configuration

and dimensions reported by Druzgalski et al 24, who simulated this problem using a

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5

uθ

d

Analytical (Ref.) PNP model

PB model DH model

V = 0.01

κ = 10

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2 2.5

uθ

d

Analytical (Ref.) PNP model

PB model DH model

V = 4

κ = 10

(a) (b)

26

second-order accurate (in time and space) finite-differences method. The distance

between the electrode and the membrane is H, and L = 6H is the length of the reservoir

in the direction perpendicular to the applied electric field (E = ΔV/H). An ion-selective

membrane is located at y = 0, which allows the flux of cationic species, but retains anionic

species. Periodic boundary conditions are imposed on the sides x = ±0.5L. We consider a

symmetric, binary electrolyte, for which z+ = -z– = z and D+ = D– = D.

FIG. 8. Geometry representing a 2D reservoir with an ion-selective membrane at y = 0 (drawing not to

scale). The reservoir width is H and the length is L = 6H. Periodic boundary conditions are imposed on the

sides, x = ±0.5L.

Two meshes were used in the numerical simulations in order to assess the

convergence with mesh refinement. Mesh M1 has 480 uniformly spaced cells in the x-

direction, and 90 cells non-uniformly distributed in the y-direction, with a minimum edge

size of H/1000 near the ion-selective membrane. Mesh M2 was obtained from M1 by

doubling the number of cells in each direction.

The following set of boundary conditions was used more details can be found in 24:

Reservoir (y = H): Ψ = ΔV, 0 np , u = 0, ci = c0, where c0 is the initial

concentration of ions in the bulk, in the absence of electric field;

Periodic boundaries (x = ±0.5L): LxLx

ΩΩ5.05.0

, where Ω represents any of

the computed variables;

Ion-selective membrane (y = 0): Ψ = 0, np given by Eq. (14), u = 0, c+ = 2c0,

F– = 0.

In order to initialize the fields, the equivalent 1D problem is solved numerically in the

absence of flow, keeping the same number of cells in the y-direction (the direction solved

x

H

L = 6H

ΔV

Ion-selective membrane

y

Reservoir

27

for), but only one single cell in the x-direction. The 1D solution is then mapped to the 2D

domain and the anionic and cationic concentration fields are locally disturbed 24 by a 1 %

random perturbation – the concentration in each cell is multiplied by a random scalar in

the range [0.99;1.01].

The dimensionless numbers governing this EDF are similar to those of the ICEO case

in the previous section: dimensionless Debye parameter,

kT

FecHzH

0

2

D

2~ ;

Schmidt number, D

Sc

; electrohydrodynamic coupling constant,

D

V

2

T ;

dimensionless potential, T

~

V

VV

, where

ez

kTV T is the thermal voltage. The following

dimensionless variables are also used in this section: 2

~

H

tDt ,

H

yy ~ ,

H

xx ~ ,

U

uu ~ ,

0

T2

)(~

c

ccc ,

0

E

)(~

c

cc and zFDc

JHJ

0

~ , where J represents the current density

and H

DU is a diffusive velocity scale. According to Druzgalski et al 24, we set χ = 0.5,

Sc = 103, κ = 103, V = 10–120 and creeping flow conditions are considered by removing

the convective term of the momentum equation. One inner-iteration is used, along with

two electrokinetic coupling iterations, and the time-step was fixed at Δt = 10-6. The

simulations were typically run until t = 1 (low voltages required more than this time to

reach steady-state).

2. Results

The time evolution of the surface-averaged current density on the membrane (Jmembrane)

is plotted in Fig. 9a for V = 10–120 (unless otherwise stated, all the results presented in

this section were computed in mesh M2). Since anions do not cross the membrane, the

surface-averaged current density is computed from (usingkT

ezD i

ii )

xcΨkT

ezDcD

L

FzJJ d

membrane

membrane,membrane n

(18)

For V ≤ 40, the current density reaches a stationary value, either due to the negligible

contribution of electroconvection (V = 10, not plotted in Fig. 9a), or due to the

establishment of quasi-steady vortices (20 ≤ V ≤ 40). For V > 40, no steady-state is

28

achieved and the current density profiles display a chaotic behavior over time for

increasing V, as shown in Fig. 9a.

FIG. 9. (a) Time evolution of the surface-averaged current density on the membrane (mesh M2). (b) Space-

and time-averaged current density on the membrane as a function of the dimensionless applied voltage. For

V > 40, the average current density is computed by averaging the profiles plotted in panel (a) in the range t

= [0.2; 1], while for V ≤ 40 the steady-state value is considered. The three regions of the J-V curve are

identified (the delimitation between regions is only approximate): A – Ohmic regime; B – limiting regime;

C – overlimiting regime.

The space- and time-averaged current density, <Jmembrane>, is presented in Fig. 9b as

a function of the dimensionless voltage. The transition from the limiting regime to the

overlimiting regime occurs at V ≈ 20. Additional simulations performed in mesh M1

showed that the transition occurs more precisely at V = 19 (results not shown), which

agrees well with other works 21, 22, 24. In general, a good agreement is observed with the

results of Druzgalski et al 24, also plotted in Fig. 9b. The major deviations, either between

our two meshes or between our results and the reference data, are observed at V = 20 and

V = 40. The first voltage is in a sensitive region of transition between the limiting and

overlimiting regimes, where a subcritical instability has been reported by several authors

21, 22, 43. The reason for the discrepancy at V = 40 is probably due to the hysteresis of the

J-V curve reported by Davidson et al 21 in the range 30 ≤ V ≤ 40, although for a geometry

with a higher aspect ratio (L/H). A vortex selection mechanism was pointed out as the

main cause for this hysteresis 21.

Fig. 10 displays the x- and time-averaged kinetic energy in the y-direction, for

different voltages (see the figure legend for details). The kinetic energy increases with the

applied voltage, which makes the vortex-conveyed charge a very plausible explanation to

(a) (b)

0

5

10

15

20

25

30

0 0.05 0.1

J mem

bra

ne

t

20 40 6080 100 120

V :

0.9 0.95 1

0

2

4

6

8

10

0 20 40 60 80 100 120<

J mem

bra

ne>

V

M1 (this work) Druz

M2 (this work) 1D solution

CBA

Druzgalski et al 24

29

the overlimiting regime. Furthermore, our profiles reproduce well those obtained by

Druzgalski et al 24, notwithstanding the chaotic behavior observed at high voltages.

FIG. 10. Space-time-averaged profiles of the kinetic energy at different voltages. At each sampling time,

the kinetic energy at each y-position was averaged along the x-direction. All the profiles collected over time

were then averaged, for each y-position. The average in time was performed in the interval t = [0.4; 1], over

20000 samples uniformly collected along that period. Symbols represent our numerical results, while lines

correspond to the data in Druzgalski et al 24 (there is no available reference data for V = 100).

The chaotic patterns of the total ionic concentration and charge density at V = 120 are

illustrated in Fig. 11 at different times. The mushroom-shaped zones of depleted fluid

formed at early times quickly disrupt into random shapes, as also observed in Karatay et

al 13. For longer times, spikes of enriched fluid inject positive charge into the EDL near

the membrane, while also removing negative charge from there. Interestingly, the

adjacent strips of opposite charge reported by Druzgalski et al 24 are also present in Fig.

11. The chaotic nature at this voltage can be further assessed and measured by spectral

analysis 13, 23, 24, a common approach in the study of turbulence. Therefore, we performed

a spatial Fourier transform of the anionic concentration in the x-direction, at different

distances from the membrane (for V = 120). The results are displayed in Fig. 12, where a

power-law decay of the energy spectra is observed over almost one decade, at y = 0.1 and

0.4. The power-law exponent in this region is approximately -2, which is close to that

obtained in Karatay et al 13 (estimated by visual inspection of their results). At y = 0.8,

close to the reservoir boundary, the region of power-law decay is shorter, probably due

to the reservoir boundary conditions and due to the finite range of action of the vortices

(the instabilities are triggered and sustained near the membrane). The spectra for the

cationic concentration are similar to those for anions (results not shown), except at y =

0.025, close to the membrane, where the spectrum for cations is much more energetic (by

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1

|u|2

/2

y

120 100 80 60 40x104 V:

. Solid lines: Druzgalski et al 24

. Symbols: M2 (this work)

30

orders of magnitude) than that for anions – the near-membrane region is essentially

depleted of anions, while cations are continuously injected on it by chaotic spikes. We

have also confirmed the existence of a chaotic behavior over time, at different positions

(results not shown).

FIG. 11. Contours of the total concentration (left panel) and dimensionless charge density (right panel, with

superimposed instantaneous streamlines) for V = 120, at different times. Note that the scale limits of the

contours do not represent necessarily the real limits of the variable represented, in order to improve visibility.

The whole domain is plotted.

A key aspect pointed out by Karatay et al 13 concerns the CPU time required to solve

a given EDF problem. In their comparative study, the authors found speedup factors as

high as one order of magnitude when using their in-house solver, in relation to the

commercial COMSOL Multiphysics® package, for a case analogue to that addressed in

this section 13. This difference is not only due to the different numerical methods used

(finite-differences vs. finite-elements), but it is also the result of a set of optimizations at

the programming level, which can be done more easily in an in-house solver, but not in a

general-purpose package because of generality reasons. In our case, using a mesh with

600 and 340 cells in the x- and y-direction (600×340×6 degrees of freedom), respectively,

at V = 120, we obtained a CPU time of approximately 2 seconds per time-step in a laptop

-0.01 0.01

0 0.004 0.008 -0.008 -0.004

t = 0.0001

t = 0.0015

t = 0.005

t = 0.99

0 1

0.25 0.5 0.75

cT ρE

E y x

31

i5-3210M processor (2.8 GHz, 3Mb cache), in a single-core run. In similar conditions,

i.e., for the same degrees of freedom and same voltage, Karatay et al 13 reported a CPU

time of approximately 1 second per time-step for their in-house solver and approximately

12 seconds per time-step using COMSOL Multiphysics®. Therefore, the OpenFOAM®

solver used in this work also offers a good compromise between computational cost and

generality (for example, in handling arbitrary geometries and grids).

FIG. 12. Energy spectra of the anionic concentration variation along the x-direction, at different y-positions,

for V = 120. In order to obtain representative data, the energy spectra were averaged over 20000 samples

uniformly collected in the period t = [0.4; 1]. The wavenumber is k = 2πx.

V. CONCLUSION

This work describes the numerical implementation of electrically-driven flow (EDF)

models in rheoTool, an open-source toolbox to simulate flows of Generalized Newtonian

and viscoelastic fluids using the finite-volume framework of OpenFOAM®.

Three EDF models were presented and discussed, including the Poisson-Nernst-

Planck model and two simplifications, the Poisson-Boltzmann model and the Debye-

Hückel model. After confirming the second-order accuracy in space and time, and the

conservativeness of the Poisson-Nernst-Planck model, the developed solver was applied

to two important EDFs: induced-charge electroosmosis around a conducting cylinder and

charge transport across an ion-selective membrane. These application cases not only

illustrated the applicability of the solver, as they also allowed to probe its accuracy and

robustness in steady/transient and smooth/chaotic flow conditions. Furthermore, the

numerical results obtained in this work increase the availability of benchmark data in non-

trivial EDFs, for which exact analytical solutions are not available or are limited to a

range of conditions.

1 10 100

E

k

y=0.025 y=0.1

y=0.4 y=0.8

-2

-310-2

10-4

10-6

10-8

10-10

32

A natural continuation of this work is the extension of EDFs to complex fluids, a

feature already available in rheoTool, but not explored in the present work. This

investigation is already under way and is left for future work. The study of EDFs in

multiphase systems is also a relevant topic to be explored. In addition, a still relatively

unexplored subject concerns EDFs with multiple species having different charge valences

and diffusivities, which can be also simulated with rheoTool.

ACKNOWLEDGEMENTS

The research leading to these results has received funding from the European

Research Council (ERC), under the European Commission “Ideas” specific programme

of the 7th Framework Programme (Grant Agreement Nº 307499). The authors thank the

relevant suggestions given by Professor Ali Mani.

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